This article is part of the series Numerical Linear Algebra in Signal Processing Applications.

Open Access Research Article

Robust Adaptive Modified Newton Algorithm for Generalized Eigendecomposition and Its Application

Jian Yang*, Feng Yang, Hong-Sheng Xi, Wei Guo and Yanmin Sheng

Author Affiliations

Laboratory of Network Communication System and Control, Department of Automation, University of Science and Technology of China, Hefei, Anhui 230027, China

For all author emails, please log on.

EURASIP Journal on Advances in Signal Processing 2007, 2007:038341 doi:10.1155/2007/38341


The electronic version of this article is the complete one and can be found online at: http://asp.eurasipjournals.com/content/2007/1/038341


Received:1 October 2006
Revisions received:13 February 2007
Accepted:16 April 2007
Published:8 August 2007

© 2007 Yang et al.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose a robust adaptive algorithm for generalized eigendecomposition problems that arise in modern signal processing applications. To that extent, the generalized eigendecomposition problem is reinterpreted as an unconstrained nonlinear optimization problem. Starting from the proposed cost function and making use of an approximation of the Hessian matrix, a robust modified Newton algorithm is derived. A rigorous analysis of its convergence properties is presented by using stochastic approximation theory. We also apply this theory to solve the signal reception problem of multicarrier DS-CDMA to illustrate its practical application. The simulation results show that the proposed algorithm has fast convergence and excellent tracking capability, which are important in a practical time-varying communication environment.

References

  1. J Lu, KN Plataniotis, AN Venetsanopoulos, Face recognition using LDA-based algorithms. IEEE Transactions on Neural Networks 14(1), 195–200 (2003). PubMed Abstract | Publisher Full Text OpenURL

  2. S Fidler, D Skočaj, A Leonardis, Combining reconstructive and discriminative subspace methods for robust classification and regression by subsampling. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(3), 337–350 (2006). PubMed Abstract | Publisher Full Text OpenURL

  3. TF Wong, TM Lok, JS Lehnert, MD Zoltowski, A linear receiver for direct-sequence spread-spectrum multiple-access systems with antenna arrays and blind adaptation. IEEE Transactions on Information Theory 44(2), 659–676 (1998). Publisher Full Text OpenURL

  4. J Yang, H Xi, F Yang, Y Zhao, Fast adaptive blind beamforming algorithm for antenna array in CDMA systems. IEEE Transactions on Vehicular Technology 55(2), 549–558 (2006). Publisher Full Text OpenURL

  5. B Yang, Projection approximation subspace tracking. IEEE Transactions on Signal Processing 43(1), 95–107 (1995). Publisher Full Text OpenURL

  6. S Ouyang, PC Ching, T Lee, Robust adaptive quasi-Newton algorithms for eigensubspace estimation. IEE Proceedings: Vision, Image and Signal Processing 150(5), 321–330 (2003). Publisher Full Text OpenURL

  7. A Hyvärinen, J Karhunen, E Oja, Independent Component Analysis (John Wiley & Sons, New York, NY, USA, 2001)

  8. GH Golub, CF VanLoan, Matrix Computations (John Hopkins University Press, Baltimore, Md, USA, 1991)

  9. G Mathew, VU Reddy, A quasi-Newton adaptive algorithm for generalized symmetric eigenvalue problem. IEEE Transactions on Signal Processing 44(10), 2413–2422 (1996). Publisher Full Text OpenURL

  10. C Chatterjee, VP Roychowdhury, J Ramos, MD Zoltowski, Self-organizing algorithms for generalized eigen-decomposition. IEEE Transactions on Neural Networks 8(6), 1518–1530 (1997). PubMed Abstract | Publisher Full Text OpenURL

  11. D Xu, JC Principe, H-C Wu, Generalized eigendecomposition with an on-line local algorithm. IEEE Signal Processing Letters 5(11), 298–301 (1998). Publisher Full Text OpenURL

  12. DR Morgan, Adaptive algorithms for solving generalized eigenvalue signal enhancement problems. Signal Processing 84(6), 957–968 (2004). Publisher Full Text OpenURL

  13. YN Rao, JC Principe, TF Wong, Fast RLS-like algorithm for generalized eigendecomposition and its applications. The Journal of VLSI Signal Processing 37(2-3), 333–344 (2004)

  14. J Yang, H Xi, F Yang, Y Zhao, RLS-based adaptive algorithms for generalized eigen-decomposition. IEEE Transactions on Signal Processing 54(4), 1177–1188 (2006)

  15. TM Lok, TF Wong, JS Lehnert, Blind adaptive signal reception for MC-CDMA systems in Rayleigh fading channels. IEEE Transactions on Communications 47(3), 464–471 (1999). Publisher Full Text OpenURL

  16. S Kondo, LB Milstein, Performance of multicarrier DS CDNA systems. IEEE Transactions on Communications 44(2), 238–246 (1996). Publisher Full Text OpenURL

  17. JG Proakis, Digital Communications (McGraw-Hill, New York, NY, USA, 1995)

  18. J Namgoong, TF Wong, JS Lehnert, Subspace multiuser detection for multicarrier DS-CDMA. IEEE Transactions on Communications 48(11), 1897–1908 (2000). Publisher Full Text OpenURL

  19. S Haykin, Adaptive Filter Theory (Prentice-Hall, Upper Saddle River, NJ, USA, 2002)

  20. L Ljung, Analysis of recursive stochastic algorithms. IEEE Transactions on Automatic Control 22(4), 551–575 (1977). Publisher Full Text OpenURL

  21. HJ Kushner, DS Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems (Springer, New York, NY, USA, 1978)

  22. DR Morgan, J Benesty, MM Sondhi, On the evaluation of estimated impulse responses. IEEE Signal Processing Letters 5(7), 174–176 (1998). Publisher Full Text OpenURL

  23. TM Lok, TF Wong, Transmitter and receiver optimization in multicarrier CDMA systems. IEEE Transactions on Communications 48(7), 1197–1207 (2000). Publisher Full Text OpenURL